Externally compatible Abelian groups of the type (2,1,0)

Krystyna Mruczek-Nasieniewska

DOI: http://dx.doi.org/10.12775/LLP.2006.014


In [4] the lattice of all subvarieties of the variety GnEx defined by so called externally compatible identities of Abelian groups together with the identity xn ≈ yn , for any n ∈ N and n ≥ 1 was described. In that paper classes of models of the type (2,1) where considered. It appears that diagrams of lattices of subvariaties defined by externally compatible identities satisfied in a given equational theory depend on the language of the considered class of algebras.

A question was asked to what extent the diagram of the lattice of subvarieties of the variety defined by externally compatible identities of a given variety will depend on changing the type of algebras. In general case, the answer to this question seems to be very complicated. In this paper we describe the variety of Abelian groups of exponent p·q, where p,q are different primes of type (2,1,0).


Abelian groups; externally compatible identities; variety

Full Text:



G. Birkhoff, On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society 31 (1935), 433–454.

W. Chromik, “On externally compatible identities of algebras”, Demonstratio Mathematica 23(2) (1990), 345–355.

K. Gajewska-Kurdziel and K. Mruczek, “On some sets of identities satisfied in Abelian groups”, Demonstratio Mathematica 35(3) (2002), 447–453.

K. Gajewska-Kurdziel and K. Mruczek-Nasieniewska, “The lattice of subvarieties of the variety defined by externally compatible identities of Abelian groups of exponent n”, Studia Logica. To appear.

I.J. Mel’nik, “Nilpotent shifts of varieties, (in Russian)”, Mat. Zametki, 14(5) (1973). English translation: Math. Notes 14 (1973), 962–966.

K. Mruczek, “On some lattice of varieties related to changes of the type”, in: K. Denecke K. and H.-J. Vogel (eds.), General Algebra and Applications, Proceedings of the 59th Workshop on General Algebra, Shaker Verlag, Aachen 2000, pp. 147–153.

K. Mruczek-Nasieniewska, “P-compatible Abelian Groups”, Logic and Logical Philosophy 14(2) (2005), 253–263.

J. Płonka, “On the subdirect product of some equational classes of algebras”, Math. Nachr. 63 (1974), 303–305.

J. Płonka, “P-compatible identities and their applications to classical algebras”, Math. Slovaca 40(1) (1990), 21–30.

A. Tarski, “Equational logic and equational theories of algebras”, in: H.A. Schmidt, K. Schütte, and H.J. Thiele (eds.), Contributions to Mathematical Logic, North Holland Publ. Co., Amsterdam 1968, pp. 275–288.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism